136 research outputs found
HAAGERUP APPROXIMATION PROPERTY VIA BIMODULES
No full textThe Haagerup approximation property (HAP) is defined for finite von Neumann algebras in such a way that the group von Neumann algebra of a discrete group has the HAP if and only if the group itself has the Haagerup property. The HAP has been studied extensively for finite von Neumann algebras and it was recently generalized to arbitrary von Neumann algebras by Caspers-Skalski and Okayasu-Tomatsu. One of the motivations behind the generalization is the fact that quantum group von Neumann algebras are often infinite even though the Haagerup property has been defined successfully for locally compact quantum groups by Daws-Fima-Skalski-White. In this paper, we fill this gap by proving that the von Neumann algebra of a locally compact quantum group with the Haagerup property has the HAP. This is new even for genuine locally compact groups
Relative Haagerup property for arbitrary von Neumann algebras
No full textWe introduce the relative Haagerup approximation property for a unital, expected inclusion of arbitrary von Neumann algebras and show that if the smaller algebra is finite then the notion only depends on the inclusion itself, and not on the choice of the conditional expectation. Several variations of the definition are shown to be equivalent in this case, and in particular the approximating maps can be chosen to be unital and preserving the reference state. The concept is then applied to amalgamated free products of von Neumann algebras and used to deduce that the standard Haagerup property for a von Neumann algebra is stable under taking free products with amalgamation over finite-dimensional subalgebras. The general results are illustrated by examples coming from q-deformed Hecke-von Neumann algebras and von Neumann algebras of quantum orthogonal groups.Green Open Access added to TU Delft Institutional Repository ‘You share, we take care!’ – Taverne project https://www.openaccess.nl/en/you-share-we-take-care Otherwise as indicated in the copyright section: the publisher is the copyright holder of this work and the author uses the Dutch legislation to make this work public.Analysi
Computational Explorations of the Thompson Group T for the Amenability Problem of F
No full textIt is a long standing open problem whether the Thompson group F is an amenable group. In this article, we show that if A, B, C denote the standard generators of Thompson group T and (Formula presented.) then (Formula presented.) Moreover, the upper bound is attained if the Thompson group F is amenable. Here, the norm of an element in the group ring (Formula presented.) is computed in (Formula presented.) via the regular representation of T. Using the “cyclic reduced” numbers (Formula presented.), and some methods from our previous article [Haagerup et al. 15] we can obtain precise lower bounds as well as good estimates of the spectral distributions of (Formula presented.) where τ is the tracial state on the group von Neumann algebra L(T). Our extensive numerical computations suggest that (Formula presented.) and, thus that F might be non-amenable. However, we can in no way rule out that (Formula presented.).</p
De ongelijkheid van Khintchine / The Khintchine inequality
No full textFirst, the Khintchine inequality and several proofs of it will be investigated, then the proof for the best constants on the Khintchine equality by Uffe Haagerup, after which a study is done on generalisatioens of the Khintchine equality and new optimal constants are found.Applied mathematicsElectrical Engineering, Mathematics and Computer Scienc
A New Look at C∗-Simplicity and the Unique Trace Property of a Group
No full textWe characterize when the reduced C∗-algebra of a non-trivial group has unique tracial state, respectively, is simple, in terms of Dixmier-type properties of the group C∗-algebra.We also give a simple proof of the recent result by Breuillard, Kalantar, Kennedy and Ozawa that the reduced C∗-algebra of a group has unique tracial state if and only if the amenable radical of the group is trivial. © Springer International Publishing Switzerland 2016
Généralisations de la propriété d'approximation de Haagerup pour les algèbres de von Neumann arbitraires
No full textThe notion of the Haagerup approximation property, originally introduced for von Neumann algebras equipped with a faithful normal tracial state, is generalised to arbitrary von Neumann algebras. We discuss two equivalent characterisations, one in term of the standard form and the other in term of the approximating maps with respect to a fixed faithful normal semifinite weight. Several stability properties, in particular regarding the crossed product construction are established and certain examples are introduced.La notion de propriété d'approximation de Haagerup, introduite à l'origine pour les algèbres de von Neumann ayant une trace finie, normale, et fidèle, est généralisée pour les algèbres de von Neumann arbitraires. Nous discutons deux caractérisations équivalentes : une du point de vue de la représentation standard et une autre du point de vue des applications linéaires approximantes liées à un poids fidèle, normal, semifini. Quelques propriétés de permanence, en particulier celles concernant les produits croisés, sont établies et certains exemples sont introduits
Simple Lie groups without the Approximation Property II
No full textWe prove that the universal covering group of Sp(2,R) does not have the Approximation Property (AP). Together with the fact that SL(3,R) does not have the AP, which was proved by Lafforgue and de la Salle, and the fact that Sp(2,R) does not have the AP, which was proved by the authors of this article, this finishes the description of the AP for connected simple Lie groups. Indeed, it follows that a connected simple Lie group has the AP if and only if its real rank is zero or one. By an adaptation of the methods we use to study the AP, we obtain results on approximation properties for noncommutative L^p-spaces associated with lattices in the universal covering group of Sp(2,R). Combining this with earlier results of Lafforgue and de la Salle and results of the second named author of this article, this gives rise to results on approximation properties of noncommutative L^p-spaces associated with lattices in any connected simple Lie group.sponsorship: The first-named author was supported by ERC Advanced Grant no. OAFPG 247321, the Danish Natural Science Research Council, and the Danish National Research Foundation through the Centre for Symmetry and Deformation (DNRF92).The second-named author was supported by the Danish National Research Foundation through the Centre for Symmetry and Deformation (DNRF92). (ERC|OAFPG 247321, Danish Natural Science Research Council, Danish National Research Foundation through the Centre for Symmetry and Deformation|DNRF92)status: Publishe
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